On a Problem Connected with Beta and Gamma Distributions by R
Total Page：16
File Type：pdf, Size：1020Kb
ON A PROBLEM CONNECTED WITH BETA AND GAMMA DISTRIBUTIONS BY R. G. LAHA(i) 1. Introduction. The random variable X is said to have a Gamma distribution G(x;0,a)if du for x > 0, (1.1) P(X = x) = G(x;0,a) = JoT(a)" 0 for x ^ 0, where 0 > 0, a > 0. Let X and Y be two independently and identically distributed random variables each having a Gamma distribution of the form (1.1). Then it is well known [1, pp. 243-244], that the random variable W = X¡iX + Y) has a Beta distribution Biw ; a, a) given by 0 for w = 0, (1.2) PiW^w) = Biw;x,x)=\ ) u"-1il-u)'-1du for0<w<l, Ío T(a)r(a) 1 for w > 1. Now we can state the converse problem as follows : Let X and Y be two independently and identically distributed random variables having a common distribution function Fix). Suppose that W = Xj{X + Y) has a Beta distribution of the form (1.2). Then the question is whether £(x) is necessarily a Gamma distribution of the form (1.1). This problem was posed by Mauldon in [9]. He also showed that the converse problem is not true in general and constructed an example of a non-Gamma distribution with this property using the solution of an integral equation which was studied by Goodspeed in [2]. In the present paper we carry out a systematic investigation of this problem. In §2, we derive some general properties possessed by this class of distribution laws Fix). In §3, we give some analytical lemmas. In §4, we make use of the lemmas and deduce a characterization of a class of dis- tribution laws having moments of a certain order. In the final section, we construct some further examples of distribution functions Fix). Received by the editors May 25,1960 and, in revised form, June 7, 1963. (!) This work was supported by the National Science Foundation through Grants NSF-G 4220 and NSF-G 9968. 287 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 288 R. G. LAHA (November 2. Some general properties of distribution laws. We first prove a lemma which is instrumental in the proof of the subsequent theorem. Lemma 2.1. Let X and Y be two independently and identically distributed random variables with a common distribution function Fix) which is continuous at the point x = 0. Suppose that the quotient V = XjY has a one-sided distribution F0iv) such that F0(u) = 0for v^O. Then X has also a one-sided distribution. Proof. As usual we assume that each of the distribution functions F(x) and F0iv) is everywhere continuous to the right so that for every v > 0, we have Foi0) - Foi - v) = Prob( - v< XjY < 0) (2.1) = JJ [F(0)-F(-!;y)]dF(y) + f [F(-t;y-0)-F(0)]dF(y). J - 00 From the conditions of the lemma we have Fo(0) = F0( — i;) = 0 for every v > 0 so that (2.1) gives fiOO (2.2a) [F(0)-F(-i;y)]dF(y) = 0, Jo (2.2b) f [F( -vy-0)- F(0)]dFiy) = 0 J-00 holding for all v > 0. Then it follows immediately from (2.2a) that either Fiy) = 0 for all y = 0 or Fiy) = 1 for all y ^ 0. It is also easy to verify that the relation (2.2b) is satisfied in either of the above cases.This completes the proof of the lemma. We next prove the following theorem: Theorem 2.1. Let X and Y be two independently and identically distributed random variables with a common distribution function Fix). Let the random variable W = XfiX + Y) follow the Beta distribution of the form (1.2). Then Fix) has the following general properties: (1) Either Fix) = Ofor x^Oor Fix) = 1for x ^ 0, (2) F(x) is absolutely continuous and has a continuous probability density function p(x) = F\x) > 0. Proof. It is easy to verify from the conditions of the theorem that the quotient V = XjY = W/il - W) has the distribution function F0(t>)given by License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 1964] BETA AND GAMMA DISTRIBUTIONS 289 r r(2a) ua_1 -du fory>0, (2.3) F0iv) = r(a)r(a) (1 +«)2« 0 for v < 0. Thus property ( 1) follows immediately from Lemma 2.1. Therefore, we can assume without any loss of generality that F(x) = 0 for x = 0, that is, the random variable X is positive. After some elementary integration we can easily derive from (2.3) the characteristic function of the distribution of In F as: Ft ¿tin v, _ Tix + it)rjx-it) He ] - Tix)Tix) • We denote the characteristic function of the distribution of In ATby ^(f) and thus obtain the basic equation T(a + ¿r)r(a- it) (2.4) <K0<K-0 = r(a)r(a) holding for all real f. Then using the elementary property of the Gamma function we can at once verify that /?,„ | i/r(f)| dt < co, that is, the characteristic function iHO is absolutely integrable. Then using the Fourier-inversion theorem we conclude that the distribution function of InX is absolutely continuous and has a con- tinuous probability density function. Then it follows easily that F(x) is also absolutely continuous and has a continuous probability density function p(x) = F'ix) > 0. 3. Some analytical lemmas. In this section we discuss some analytical lemmas which are instrumental in the proofs of the subsequent theorems. Lemma 3.1. Let /(z) be an analytic characteristic function which is regular in a certain horizontal strip of maximum width containing the real axis. Then the purely imaginary points on the boundary of this strip of regularity are singular points offz). This lemma is due to Levy [5]. A proof of this lemma is given by Lukacs in [8]. Lemma 3.2. Let /(z) be a decomposable characteristic function which is regular in a strip — ß < Imz < ß iß >0) of the complex z-plane. LetfAz) be a factor offiz). Then the characteristic function fxiz) is also regular at least in the same strip. This lemma on the factorization of analytic characteristic functions is due toRaikov[10]. Lemma 3.3. Let Riz) be an entire function of order unity having only purely imaginary zeros and R(0) = 1. Then its reciprocal 1/R(z) is always a charac- teristic function. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 290 R. G. LAHA [November The proof of this lemma is given in [6, p. 140]. Lemma 3.4. Letfz) be an analytic characteristic function and suppose that fo(z) = [/(z) +/( - z)]/2 andfiz) = [/(z) -/( - z)]/2 are respectively the even and odd parts offiz). Let p,p0 and px denote respectively the radii of convergence of Maclaurin series for /(z),/0(z) andfxiz). Then p = p0 = px. This lemma has been proved by the author in [3]. Lemma 3.5. Let fz) be the characteristic function of an infinitely divisible distribution which is regular in a certain horizontal strip. Then /(z) has no zeros inside its strip of regularity and therefore </>(z)= ln/(z) is defined and regular in the same strip. Moreover, the function 0(z) = <b"iz)l<b"i0) is also a characteristic function which is regular in the same strip. The proof of this lemma is given in [8]. Lemma 3.6. Let p(x) be a continuous non-negative function of the real variable x. Let the integral J^ x"p(x) dx (y reaO exist for all v in the interval 0 < v < V iV> 0).Then the integral Iiz)= f0°°x~lzp(x)dx as a function of the complex variable z is regular in the strip 0 < Imz < Vof the upper half-plane. This lemma has been proved by the author in [4]. As a special case of this lemma, we note that if the integral J" x"pix)dx exists for all real v > 0, then the integral I{z) is regular throughout the upper half-plane Imz > 0. Lemma 3.7. Let the distribution function F(x) of a positive random variable X be absolutely continuous and have a continuous probability density function p(x) = F'(x) > 0. Let Fix) possess finite absolute moments up to a certain order ô (á > 0 not necessarily an integer). Let rpit) = Eie"inX) denote the characteristic function of the distribution oflnX. Then the function ¡¡/i-z) = Eie-izlnX) considered as a function of the complex variable z is regular in the strip 0 < Imz < 5 + £ (s > 0 may be a suficiently small number). Conversely if the function \pi — z) is regular in the strip 0<Imz<<5+£ (s > 0 arbitrarily small), then F(x) has finite absolute moments up to order <5. Proof. Since the distribution function F(x) has finite absolute moments up to order <5,the integral f™x"pix)dx exists for all real v in the interval 0 < v < 5 + e (where s may be an arbitrarily small positive number). We further note that iK - z) = £(e"izInX ) = r x~izpix)dx. Then the proof follows at once from Lemma 3.6. The proof of the converse statement is immediate, License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 1964] BETA AND GAMMA DISTRIBUTIONS 291 As a special case of Lemma 3.7, it follows that if F(x) has finite moments of all orders, then the function i/r( - z) is regular throughout the upper half-plane lmz>0.